One-dimensional discrete Gaussian Markov processes: Harmonic decomposition of invariant boundary conditions

Abstract: We study invariant boundary conditions for one dimensional discrete Gaussian Markov processes, basic toy models of spatial Markov processes in statistical mechanics. More precisely, we give a decomposition of boundary objects in a non trivial basis from the study of a meromorphic matrix-valued function $\Phi$ (inherent to the model) and its singularities. This provides a simple algorithm for the explicit computation of invariant measures. As an application, we give an "eigen" version of Szeg\H{o} limit theorem for matrix valued trigonometric polynomials.